# Biased Estimators on Quotient Spaces

## Abstract

Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the *interaction between statistics and geometry*? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes -and more generally on observations belonging to quotient spaces- have been studied since the 1980’s. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.

## Keywords

Quotient Space Orbit Type Riemannian Foliation Isometric Action Singular Orbit## References

- 1.Alekseevsky, D., Kriegl, A., Losik, M., Michor, P.W.: The Riemannian geometry of orbit spaces. The metric, geodesics, and integrable systems (2001)Google Scholar
- 2.Allassonnière, S., Amit, Y., Trouvé, A.: Towards a coherent statistical framework for dense deformable template estimation. J. Roy. Stat. Soc.
**69**(1), 3–29 (2007)MathSciNetGoogle Scholar - 3.Bauer, M., Bruveris, M., Michor, P.: Overview of the geometries of shape spaces and diffeomorphism groups. J. Math. Imaging Vis.
**50**(1–2), 60–97 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Grenander, U., Miller, M.: Computational anatomy: an emerging discipline. Q. Appl. Math.
**LVI**(4), 617–694 (1998)MathSciNetzbMATHGoogle Scholar - 5.Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: geodesic principal component analysis for riemannian manifolds modulo lie group actions. Stat. Sin.
**20**, 1–100 (2010)MathSciNetzbMATHGoogle Scholar - 6.Joshi, S., Kaziska, D., Srivastava, A., Mio, W.: Riemannian structures on shape spaces: a framework for statistical inferences. In: Krim, H., Yezzi Jr., A. (eds.) Statistics and Analysis of Shapes, pp. 313–333. Birkhäuser, Boston (2006)CrossRefGoogle Scholar
- 7.Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc.
**16**(2), 81–121 (1984)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Kendall, D.G.: A survey of the statistical theory of shape. Stat. Sci.
**4**(2), 87–99 (1989)MathSciNetCrossRefzbMATHGoogle Scholar - 9.Kurtek, S.A., Srivastava, A., Wu, W.: Signal estimation under random time-warpings and nonlinear signal alignment. In: Advances in Neural Information Processing Systems 24, 675–683 (2011)Google Scholar
- 10.Le, H., Kendall, D.G.: The Riemannian structure of Euclidean shape spaces: a novel environment for statistics. Ann. Stat.
**21**(3), 1225–1271 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Lytchak, A., Thorbergsson, G.: Curvature explosion in quotients and applications. J. Differ. Geom.
**85**(1), 117–140 (2010)MathSciNetzbMATHGoogle Scholar - 12.Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis.
**25**(1), 127–154 (2006)MathSciNetCrossRefGoogle Scholar - 13.Postnikov, M.: Geometry VI: riemannian geometry. Encyclopaedia of Mathematical Sciences. Springer (2001)Google Scholar
- 14.Small, C.: A survey of the statistical theory of shape. Stat. Sci. (4), 105–108 (1989). The Institute of Mathematical StatisticsGoogle Scholar